Acoustic interrogation of subsurface features tends to be limited by the size and power of practical sources, and in practice, the output of down hole acoustic transducers is limited by the power transmission capabilities of the wireline cable. High frequency signals have a relatively short penetration distance, while low frequency signals generally require large sources, clamped to the borehole wall, to maximize energy transfer to the formation and minimize unwanted signals within the well bore. Currently, acoustic borehole tools are designed with acoustic sources in the borehole to detect returning acoustic waves that are propagating along the borehole walls or scattered by inhomogeneities of linear properties of rock formations surrounding the borehole. U.S. Pat. No. 7,301,852 to Leggett, III et al. discloses a Logging While Drilling tool, designed to detect rock formation boundaries. The tool uses two acoustic source arrays emitting two acoustic waves from a borehole that generate a third wave by non-linear mixing in the rock formation surrounding the borehole at the location of intersection of the acoustic waves. The third wave continues forward and interacts linearly with heterogeneities in the subsurface properties. The third wave is scattered by the heterogeneities in the subsurface properties, and the scattered signal is detected by sensors in the logging tool. U.S. Pat. No. 7,301,852 does not discuss detecting the third wave directly but rather the signal that is scattered by the heterogeneities in the rock formation. U.S. Pat. No. 7,301,852 merely uses the resultant scattered wave to detect rock formation boundaries.
Attempts have been made to characterize the non-linear properties of a formation in the area of oil and gas prospecting from boreholes, but each has its own limitations. For example, U.S. Pat. No. 5,521,882 to D'Angelo et al. discloses an acoustic tool designed to record with pressure receivers non-linear waves generated by non-linear mixing of two waves. The non-linear waves propagate along the borehole wall with limited penetration into the surrounding rock formation and refract back into the well bore fluid. The indication of non-linearity is utilized to provide an indication of the relative consolidation of the formation surrounding the borehole. U.S. Pat. No. 5,521,882 does not discuss measuring non-linear characteristics of a rock formation away from the borehole. U.S. Pat. No. 6,175,536 by Khan discloses a method to estimate the degree of non-linearity of earth formations from spectral analysis of seismic signals transmitted into the earth formations from a first borehole and received in a second borehole. The method in U.S. Pat. No. 6,175,536 determines from the spectral analysis the presence of a frequency at a receiver located at the second borehole representing a sum or a difference of two selected frequencies of the transmitted seismic signals generated by two sources located at the first borehole. U.S. Pat. No. 6,175,536 does not discuss measuring non-linear characteristics of a rock formation in a remote region of a borehole where the receiver and the sources are located in one borehole.
In light of these prior attempts, there is a need for a system and method for characterizing non-linear properties in a remote region from a borehole.